0.1 By Month

0.2 CCAMLR Units

0.3 Distance

0.4 Time

0.5 Velocity

0.6 Angles

1 Correlated random walk

Process Model

\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]

1.1 Parameters

For each individual:

\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]

For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]

1.2 Behavioral States

\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]

\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]

\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]

With the probability of switching states:

\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]

\[\phi_{foraging} = 1 - \phi_{traveling} \]

1.3 Continious tracks

The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.

How did the filter change the extent of tracks?

sink(“Bayesian/Multi_RW.jags”) cat(" model{

#Constants
pi <- 3.141592653589

##argos observation error##
argos_prec[1:2,1:2] <- inverse(argos_sigma*argos_cov[,])

#Constructing the covariance matrix
argos_cov[1,1] <- 1
argos_cov[1,2] <- sqrt(argos_alpha) * rho
argos_cov[2,1] <- sqrt(argos_alpha) * rho
argos_cov[2,2] <- argos_alpha

for(i in 1:ind){
for(g in 1:tracks[i]){

## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec)

#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)

###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs

#Process Model for movement
for(t in 2:(steps[i,g]-1)){

#Behavioral State at time T
logit(phi[i,g,t,1]) <- alpha_mu[state[i,g,t-1]] 
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])

#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])

#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])

#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}

#Final behavior state
logit(phi[i,g,steps[i,g],1]) <- alpha_mu[state[i,g,steps[i,g]-1]] 
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])

##  Measurement equation - irregular observations
# loops over regular time intervals (t)    

for(t in 2:steps[i,g]){

# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){ 
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]

#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec)
}
}
}
}
###Priors###

#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)

##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)

# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi

# prior for theta in 'foraging state'    
theta[2] <- (tmp[2] * pi * 2)

##Move persistance
# prior for gamma (autocorrelation parameter)
#from jonsen 2016
gamma[1] ~ dbeta(5,2)   ## gamma for state 1: traveling
dev ~ dbeta(1,1)            ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2] <- gamma[1] * dev      ## gamma for state 1


##Behavioral States

#Hierarchical structure across motnhs
#Intercepts
alpha_mu[1] ~ dnorm(0,0.386)
alpha_mu[2] ~ dnorm(0,0.386)

#Variance
alpha_tau[1] ~ dt(0,1,1)I(0,)
alpha_tau[2] ~ dt(0,1,1)I(0,)

#Probability of behavior switching 
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]

##Argos priors##
#longitudinal argos error
argos_sigma ~ dunif(0,10)

#latitidunal argos error
argos_alpha~dunif(0,10)

#correlation in argos error
rho ~ dunif(-1, 1)


}"
,fill=TRUE)

sink()

##      user    system   elapsed 
##   421.377     3.302 45427.143

1.4 Chains

##                         Type       Size     PrettySize  Rows Columns
## jagM          rjags.parallel 1522200512   [1] "1.4 Gb"     6      NA
## data                    list  106341512 [1] "101.4 Mb"     9      NA
## argos                  array   69920064  [1] "66.7 Mb"    41      15
## obs                    array   69920064  [1] "66.7 Mb"    41      15
## j                      array   34968200  [1] "33.3 Mb"    41      15
## mdat              data.frame   16339200  [1] "15.6 Mb" 49859      47
## mxy               grouped_df   15777992    [1] "15 Mb" 48723      54
## sxy                     list   15726544    [1] "15 Mb"   123      NA
## d     SpatialPointsDataFrame   14373352  [1] "13.7 Mb" 49859      49
## oxy               data.frame   13572520  [1] "12.9 Mb" 49859      49
##             used   (Mb) gc trigger   (Mb)  max used   (Mb)
## Ncells   1743990   93.2    3205452  171.2   3205452  171.2
## Vcells 266741616 2035.1  550872977 4202.9 476553410 3635.9

1.4.1 Compare to priors

1.5 Parameter Summary

##   parameter         par        mean       lower       upper
## 1  alpha_mu alpha_mu[1] -0.52156167 -0.91448547 -0.14420363
## 2  alpha_mu alpha_mu[2] -1.44659642 -1.72554890 -1.13807985
## 3     gamma    gamma[1]  0.98811826  0.96991406  0.99784163
## 4     gamma    gamma[2]  0.22477019  0.19500863  0.25394408
## 5     theta    theta[1] -0.02917987 -0.06758932  0.01336358
## 6     theta    theta[2]  3.19744268  0.15000598  6.22479804

2 Behavioral Prediction

2.1 Spatial Prediction

Compared to CMLRR regions

2.1.1 Per Animal

2.2 Autocorrelation in behavior

2.3 Behavioral description

2.4 Location of Behavior

3 Overlap with Krill Fishery

4 Time spent in grid cell

4.1 Traveling

##                           Type      Size     PrettySize    Rows Columns
## pc                      tbl_df 269622488 [1] "257.1 Mb" 5172000      10
## data                      list 106341512 [1] "101.4 Mb"       9      NA
## argos                    array  69920064  [1] "66.7 Mb"      41      15
## obs                      array  69920064  [1] "66.7 Mb"      41      15
## ssm   SpatialPolygonsDataFrame  60340504  [1] "57.5 Mb"      17       6
## j                        array  34968200  [1] "33.3 Mb"      41      15
## csmm  SpatialPolygonsDataFrame  28125776  [1] "26.8 Mb"       8       6
## mdat                data.frame  16339200  [1] "15.6 Mb"   49859      47
## sxy                       list  15206392  [1] "14.5 Mb"      41      NA
## mxy                 data.frame  15202512  [1] "14.5 Mb"   46847      57
##            used  (Mb) gc trigger   (Mb)  max used   (Mb)
## Ncells  2521144 134.7    5497235  293.6   5684620  303.6
## Vcells 65505334 499.8  180510056 1377.2 549704172 4194.0